Theorem rabssab 3690

Description: A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
rabssab	|- { x e. A | ph } C_ { x | ph }

Proof of Theorem rabssab

Step	Hyp	Ref	Expression
1		df-rab 2921	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		simpr 477	. . 3 |- ( ( x e. A /\ ph ) -> ph )
3	2	ss2abi 3674	. 2 |- { x | ( x e. A /\ ph ) } C_ { x | ph }
4	1, 3	eqsstri 3635	1 |- { x e. A | ph } C_ { x | ph }

Syntax hints:   /\ wa 384   e. wcel 1990   {cab 2608   {crab 2916   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-in 3581  df-ss 3588

This theorem is referenced by:  epse  5097  riotasbc  6626  toponsspwpw  20726  dmtopon  20727  aannenlem2  24084  aalioulem2  24088  ballotlemfmpn  30556  rencldnfilem  37384  rababg  37879